On the Jacobi metric for a general Lagrangian system

Abstract

An explicit expression for the Jacobi metric for a general Lagrangian system is obtained as a series expansion in the square root of the kinetic energy of the system and the corresponding geodesics are described in terms of an appropriate non-linear connection and the associated curvature. In the limit of low kinetic energies the trajectories of motion of any Lagrangian system are very well approximated by the geodesics of an energy dependent Randers metric or, equivalently, by the paths in configuration space of a representative point subject to electromagnetic- and gravitational-like force fields. For higher kinetic energy values the trajectories of motion are instead the geodesics of a general energy dependent Finsler metric, corresponding also to the paths in configuration space of a representative point subject to a hierarchy of a potentially infinite number of covariant force fields that generalise the electromagnetic and gravitational ones. Some general implications of these findings are discussed.

arXiv comment16 pages, no figures

References in corpus (6)

• Finsler geometric extension of Einstein gravity
• Causal structure and electrodynamics on Finsler spacetimes
• The Jacobi-metric for timelike geodesics in static spacetimes
• Finsler geometry as a model for relativistic gravity
• The connections of pseudo-Finsler spaces
• Finsler geodesics of Lagrangian systems through Routh reduction

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• Electromagnetic and gravitational interactions from Lagrangian mechanics
• Deeper understandings of the gauge theory for the first order inhomogeneous linear elasticity